SCAN2016,Uppsala,2016 L.JaulinandB.Zerr Secureazonewithrobots

(1)

Thick sets

Secure a zone with robots

L. Jaulin and B. Zerr

ENSTA-Bretagne, UBO, Lab-STICC

SCAN 2016, Uppsala, 2016

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Thick sets

Secure a zone

L. Jaulin and B. Zerr Secure a zone with robots

(3)

Thick sets

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Thick sets

Bay of Biscay 220 000 km ²

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Thick sets

An intruder

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Thick sets

Several robots R 1 , . . . , R n at positions a ₁ , . . . , a _n are moving in the ocean.

If the intruder is in the visibility zone of one robot, it is detected.

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Thick sets

Complementary approach

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Thick sets

We assume that a virtual intruder exists inside G .

We localize it with a set-membership observer inside X (t ).

The secure zone corresponds to the complementary of X (t).

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Thick sets

Assumptions

The intruder satisfies

x ˙ ∈ F (x(t)).

Each robot R i has the visibility zone g _a ⁻¹

_i

([0,d _i ]) where d i is

the scope.

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Thick sets

Theorem. An (undetected) intruder has a state vector x(t) inside the set

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt)) ∩ ^\

i

g _a ⁻¹

i

(t) ([d i (t), ∞]),

where X (0) = G . The secure zone is

S (t) = proj _world ( X (t)).

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Thick sets

Set G in white

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Thick sets

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Thick sets

green: ^S _i g _a ⁻¹

i

(t) ([0, d i (t )])

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Thick sets

Blue: G ∩ ^T _i g _a ⁻¹

i

(t) ([d i (t), ∞])

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Thick sets

Blue:

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt ))) ∩ ^T _i g _a ⁻¹ ([d i (t),∞]).

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Thick sets

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Thick sets

Strategy of the ellipse

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Thick sets

Video : https://youtu.be/rNcDW6npLfE

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Thick sets

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Thick sets

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Thick sets

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Thick sets

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Thick sets

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Thick sets

If a _i (t) ∈ [a _i ] (t), the thick observer is

JXK (t) = G ∩ ( JXK (t − dt) + dt · F ( JXK (t − dt ))) ∩ ^\

i

g _[a ⁻¹

i

](t) ( J [d _i (t)],∞ K ).

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Thick sets

Non causal secure zone

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Thick sets

Idea: Take into account the future. If H is the set-membership flow, we have

X (t) = G ∩ ( X (t − dt ) + dt · F ( X (t − dt)))

∩ ^T _t

₁

_≥t H t−t

₁

T

i g _a ⁻¹

i

(t

1

SCAN2016,Uppsala,2016 L.JaulinandB.Zerr Secureazonewithrobots

Secure a zone with robots

L. Jaulin and B. Zerr

ENSTA-Bretagne, UBO, Lab-STICC

SCAN 2016, Uppsala, 2016

Secure a zone

Bay of Biscay 220 000 km 2

An intruder

Several robots R 1 , . . . , R n at positions a 1 , . . . , a n are moving in the ocean.

If the intruder is in the visibility zone of one robot, it is detected.

Complementary approach

We assume that a virtual intruder exists inside G .

We localize it with a set-membership observer inside X (t ).

The secure zone corresponds to the complementary of X (t).

Assumptions

The intruder satisfies

x ˙ ∈ F (x(t)).

Each robot R i has the visibility zone g a −1

([0,d i ]) where d i is

the scope.

Theorem. An (undetected) intruder has a state vector x(t) inside the set

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt)) ∩ \

i

g a −1

(t) ([d i (t), ∞]),

where X (0) = G . The secure zone is

S (t) = proj world ( X (t)).

Set G in white

green: S i g a −1

(t) ([0, d i (t )])

Blue: G ∩ T i g a −1

(t) ([d i (t), ∞])

Blue:

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt ))) ∩ T i g a −1 ([d i (t),∞]).

Strategy of the ellipse

Video : https://youtu.be/rNcDW6npLfE

Thick sets

If a i (t) ∈ [a i ] (t), the thick observer is

JXK (t) = G ∩ ( JXK (t − dt) + dt · F ( JXK (t − dt ))) ∩ \

i

g [a −1

](t) ( J [d i (t)],∞ K ).

Non causal secure zone

Idea: Take into account the future. If H is the set-membership flow, we have

X (t) = G ∩ ( X (t − dt ) + dt · F ( X (t − dt)))

∩ T t

≥t H t−t

T

i g a −1

(t

) ([d i (t 1 ),∞])

.

Bay of Biscay 220 000 km ²

Several robots R 1 , . . . , R n at positions a ₁ , . . . , a _n are moving in the ocean.

Each robot R i has the visibility zone g _a ⁻¹

([0,d _i ]) where d i is

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt)) ∩ ^\

g _a ⁻¹

S (t) = proj _world ( X (t)).

green: ^S _i g _a ⁻¹

Blue: G ∩ ^T _i g _a ⁻¹

X(t) = G ∩ (X(t − dt ) + dt · F(X(t − dt ))) ∩ ^T _i g _a ⁻¹ ([d i (t),∞]).

If a _i (t) ∈ [a _i ] (t), the thick observer is

JXK (t) = G ∩ ( JXK (t − dt) + dt · F ( JXK (t − dt ))) ∩ ^\

g _[a ⁻¹

](t) ( J [d _i (t)],∞ K ).

∩ ^T _t

_≥t H t−t

i g _a ⁻¹